Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
18
votes
9answers
1k views
$i^2$ why is it $-1$ when you can show it is $1$?
We know $$i^2=-1 $$then why does this happen?
$$
i^2 = \sqrt{-1}\times\sqrt{-1}
$$
$$
=\sqrt{-1\times-1}
$$
$$
=\sqrt{1}
$$
$$
= 1
$$
EDIT: I see this has been dealt with before but at least with ...
11
votes
2answers
946 views
-1 is not 1, so where is the mistake?
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\ \\
\frac1{\sqrt{-1}} &= \frac1i \\ \\
\frac{\sqrt1}{\sqrt{-1}} ...
24
votes
7answers
2k views
Do odd imaginary numbers exist?
Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
205
votes
33answers
17k views
Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
20
votes
9answers
7k views
How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ?
thanks.
24
votes
3answers
1k views
28
votes
19answers
2k views
Interesting results easily achieved using complex numbers
I was just looking at a calculus textbook preparing my class for next week on complex numbers. I found it interesting to see as an exercise a way to calculate the usual freshman calculus integrals ...
7
votes
7answers
1k views
What's the thing with $\sqrt{-1} = i$
What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is ...
1
vote
2answers
245 views
Division of Complex Numbers
Ahlfors says that once the existence of the quotient $\frac{a}{b}$ has been proven, its value can be found by calculating $\frac{a}{b} \cdot \frac{\bar b}{\bar b}$. Why doesn't this manipulation show ...
17
votes
8answers
9k views
what is the square root of i?
If i is the square root of -1, is the square root of i imaginary? Is it used or considered often in mathematics? How is it notated?
12
votes
4answers
6k views
How do I get the square root of a complex number?
If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
12
votes
1answer
1k views
Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
Using n_th root of unity
$(e^{\frac{2ki\pi}{n}})^{n} = 1$
Prove that
$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
5
votes
4answers
1k views
Non-integer powers of negative numbers
Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically:
Can we define fractional powers such as (-2)^-1.5?
Can we define ...
12
votes
4answers
384 views
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
I was reviewing some matrices and found this interesting
if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
5
votes
1answer
707 views
Simple Complex Number Problem: 1 = -1 [duplicate]
Possible Duplicate:
-1 is not 1, so where is the mistake?
I'm trying to understand the exact point of failure in the following reasoning:
$1 = \sqrt{1} = \sqrt{(-1)(-1)} = ...
10
votes
4answers
426 views
Which step in this process allows me to erroneously conclude that $i = 1$
I was playing around with imaginary numbers and exponents and came up with this:
$$ i = \sqrt{-1} $$
$$ \sqrt{-1} = (-1)^{1/2} $$
$$ (-1)^{1/2} = (-1)^{2/4} $$
$$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...
5
votes
3answers
517 views
How to raise a complex number to the power of another complex number?
How do I calculate the outcome of taking one complex number to the power of another, ie $\displaystyle {(a + bi)}^{(c + di)}$?
6
votes
5answers
652 views
Understanding imaginary exponents
Greetings!
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to ...
4
votes
3answers
292 views
Do the real numbers and the complex numbers have the same cardinality?
So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid.
Can the approach be extended to say that the set of complex numbers has ...
0
votes
5answers
296 views
Simple doubt about complex numbers
The question itself is simple, but I'm weak in math, and I'm training a lot every day to be the best I can, so, working on complex numbers, I got stuck on a simple multiplication:
$\sqrt{3i} * 2i$
...
9
votes
8answers
535 views
For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$
If $|z|=1$, show that: $$\mathrm{Re}\left(\frac{1 - z}{1 + z}\right) = 0$$
I reasoned that for $z = x + iy$, $\sqrt{x^2 + y^2} = 1\implies x^2 + y ^2 = 1$ and figured the real part would be:
...
4
votes
5answers
527 views
How to combine complex powers?
Regarding this thread,
it is not possible to combine complex powers in the usual way:
$$ (x^y)^z = x^{yz} $$
There was mention of multi-valued functions, is there some theory that makes this all ...
9
votes
5answers
814 views
How to compute $\sqrt{i + 1}$ [duplicate]
Possible Duplicate:
How do I get the square root of a complex number?
I'm currently playing with complex numbers and I realized that I don't understand how to compute $\sqrt{i + 1}$. My ...
4
votes
2answers
180 views
General question on relation between infinite series and complex numbers
This is a strictly preliminary question.
I hope to elicit some discussion/s which will lead to a more acceptable form for the question on this site.
I'm trying to understand how the study of the ...
23
votes
4answers
2k views
Prove that $i^i$ is a real number
According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
9
votes
2answers
462 views
Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$
1)
Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$.
Prove: Is it true that $a \in \mathbb{Q}$?
2) Suppose $a \in \mathbb{C}$, ...
4
votes
3answers
355 views
Comparing real and complex numbers
If I'm correct, a complex number can be interpreted as a set in the following manner:
$$
\forall x, y \in \mathbb{R}, x + yi = \{(x,\ y)\}.\ \mathbf{(1)}
$$
My question is, is it technically ...
4
votes
3answers
382 views
How can you find the complex roots of i?
A variation of the Root of Unity problem.
I want to find all possible answers to this:
$$z^n = i$$
Where $$i^2 = -1$$
2
votes
1answer
116 views
Prove the following equation of complex power series.
Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have
$$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$
$$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$
My guess ...
3
votes
2answers
90 views
Linear Algebra of Symmetric Sums
I believe the following is true, but I do not know how to prove it.
Given two sets of distinct non-zero complex numbers $z_1, \dots , z_n$ and $w_1, \dots , w_n$, if
$$ \begin{bmatrix}
z_{1} ...
2
votes
3answers
577 views
Complex Exponents
What does it mean to raise a number to a complex exponent, and why? A lot of the explanations that I've seen involve e, why is this?
I'm looking for an intuitive answer describing to how ...
0
votes
1answer
120 views
For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?
For example, when $\alpha = 2$, $\ln(z^{2}) \neq 2\ln(z)$, because argument z is determined up to constant $2 \pi k$. So
$$
\ln(z^{2}) = \ln(z) + \ln(z) = \ln(z_{k_{1}}) + \ln(z_{k_{2}}) \neq ...
13
votes
7answers
1k views
How can one intuitively think about quaternions?
Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the ...
16
votes
4answers
587 views
Mandelbrot fractal: How is it possible?
I'm a programmer and have recently played around a bit with rendering Mandelbrot fractals / zooming into them.
What I can't grasp: How can such infinite, complex shapes come out of somewhat 10 lines ...
14
votes
2answers
800 views
De Moivre's Theorem. Motivation and origins.
I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous
$$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$
but ...
9
votes
2answers
939 views
Do “imaginary” and “complex” angles exist?
During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting:
$ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $
$ \sin ...
4
votes
6answers
891 views
Plot $|z - i| + |z + i| = 16$ on the complex plane
Plot $|z - i| + |z + i| = 16$ on the complex plane
Conceptually I can see what is going on. I am going to be drawing the set of points who's combine distance between $i$ and $-i = 16$, which will ...
13
votes
9answers
1k views
Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]
Possible Duplicate:
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
On Wikipedia, it says that:
Matrix ...
11
votes
5answers
704 views
What's bad about calling $i$ “the square root of -1”?
I vaguely recall a teacher telling me that he dislikes introducing the imaginary unit $i$ as "the square root of $-1$", but I can't remember why. Is there a lack of rigour in the statement, or is it a ...
9
votes
7answers
1k views
Is there a formula for $(1+i)^n+(1-i)^n$?
I'm wondering if there is a formula for the value of $(1+i)^n+(1-i)^n$?
I calculated the first terms starting with $n=1$ to be, in order, $2$, $0$, $-4$, $-8$, $-8$, $0$, $16$, $\dots$
So it seems ...
4
votes
3answers
606 views
Complex conjugate of $z$ without knowing $z=x+i y$
Is it possible to determine (and if so, how) the complex conjugate $\bar{z}$ of $z$, if you don't already know that $z = x + i y$?
I think you can use $\log(z)$ to get the angle, and therefore the ...
-3
votes
4answers
206 views
How one should solve $x^2+\frac{81x^2}{(x+9)^2}=40$
As in the title,
Please help me solve $x^2+\frac{81x^2}{(x+9)^2}=40$
Thanks.
11
votes
2answers
395 views
Inequality with Complex Numbers
Consider the following problem:
Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that
$$\left|\sum_{j\in J} z_j\right|\ge ...
5
votes
2answers
193 views
Primitive roots of unity
I am trying to show that,
If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then
$$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} ...
3
votes
1answer
64 views
Does anyone know any resources for Quaternions for truly understanding them?
I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
3
votes
1answer
209 views
Sum of every $k$th binomial coefficient.
It is widely known that $$\sum_{m=0}^{n} {n\choose m} = 2^n$$
and that $$\sum_{m=0}^{\lfloor\frac{n}{2}\rfloor}{n\choose 2m} = 2^{n-1}$$ Both results can be proven by exploting the nature of the roots ...
2
votes
5answers
236 views
How does $e^{i x}$ produce rotation around the imaginary unit circle?
Euler' formula states that: $e^{i x} = \cos(x) + i \sin(x)$
I can see from the MacLaurin Expansion that this is indeed true, however, I don't intuitively understand how raising $e^{i x}$ power ...
2
votes
5answers
256 views
Complex number: calculate $(1 + i)^n$.
I have to solve the following complex number exercise: calculate $(1 + i)^n\forall n\in\mathbb{N}$ giving the result in $a + ib$ notation.
Basically what I have done is calculate $(1 + i)^n$ for some ...
6
votes
3answers
174 views
4 dimensional numbers
I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
5
votes
2answers
346 views
Proof of an inequality about $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$
I've encountered an inequality pertaining to the following expression:
$\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$, where $z$ is a complex number.
After writing $z$ as $x + iy$ we have ...
